Reproducing kernels and conformal mappings in \(\mathbb{R}^n\)
In this paper the author considers the theory of reproducing kernels for spaces of Clifford-valued functions. In classical complex function theory reproducing kernels have a wide range of applications and relations given by Riemann's mapping theorem. Because in higher dimensions the only conformal mappings are Möbius transformations the possibilities are limited. The main result of the paper is that reproducing kernels are solutions of a minimum problem, which is a non-trivial extension of the analogous property for real and complex valued functions. Further the author deals with the Szegö and the Bergman kernel. The conformal invariance of the Szegö kernel under Möbius transformations is proved and relations between Szegö and Bergman kernel are given. These kernels are explicitly calculated for the unit ball and the half space.
scientific article; zbMATH DE number 1218581